Related papers: Hypergraph regularity and higher arity VC-dimensio…
Our main result is that every graph $G$ on $n\ge 10^4r^3$ vertices with minimum degree $\delta(G) \ge (1 - 1 / 10^4 r^{3/2} ) n$ has a fractional $K_r$-decomposition. Combining this result with recent work of Barber, K\"uhn, Lo and Osthus…
Is detecting a $k$-clique in $k$-partite regular (hyper-)graphs as hard as in the general case? Intuition suggests yes, but proving this -- especially for hypergraphs -- poses notable challenges. Concretely, we consider a strong notion of…
We introduce the problem Partial VC Dimension that asks, given a hypergraph $H=(X,E)$ and integers $k$ and $\ell$, whether one can select a set $C\subseteq X$ of $k$ vertices of $H$ such that the set $\{e\cap C, e\in E\}$ of distinct…
For $k\ge 3$ and $\epsilon>0$, let $H$ be a $k$-partite $k$-graph with parts $V_1,\dots, V_k$ each of size $n$, where $n$ is sufficiently large. Assume that for each $i\in [k]$, every $(k-1)$-set in $\prod_{j\in [k]\setminus \{i\}} V_i$…
The Vapnik-\v{C}ervonenkis dimension is a complexity measure of set-systems, or hypergraphs. Its application to graphs is usually done by considering the sets of neighborhoods of the vertices (cf. Alon et al. (2006) and Chepoi, Estellon,…
A $k$-graph (or $k$-uniform hypergraph) $H$ is uniformly dense if the edge distribution of $H$ is uniformly dense with respect to every large collection of $k$-vertex cliques induced by sets of $(k-2)$-tuples. Reiher, R\"odl and Schacht…
We describe a new family of $k$-uniform hypergraphs with independent random edges. The hypergraphs have a high probability of being peelable, i.e. to admit no sub-hypergraph of minimum degree $2$, even when the edge density (number of edges…
A packing of two $k$-uniform hypergraphs $H_1$ and $H_2$ is a set $\{H_1', H_2'\}$ of edge-disjoint sub-hypergraphs of the complete $k$-uniform hypergraph $K_n^{(k)}$ such that $H_1'\cong H_1$ and $H_2'\cong H_2$. Whilst the problem of…
Given $k$-uniform hypergraphs $G$ and $H$ on $n$ vertices with densities $p$ and $q$, their relative discrepancy is defined as $\hbox{disc}(G,H)=\max\big||E(G')\cap E(H')|-pq\binom{n}{k}\big|$, where the maximum ranges over all pairs…
A sequence of $k$-uniform hypergraphs $H_1, H_2, \dots$ is convergent if the sequence of homomorphism densities $t(F, H_1), t(F, H_2), \dots$ converges for every $k$-uniform hypergraph $F$. For graphs, Lov\'asz and Szegedy showed that every…
Let $H_k^r$ denote an $r$-uniform hypergraph with $k$ edges and $r+1$ vertices, where $k \leq r+1$ (it is easy to see that such a hypergraph is unique up to isomorphism). The known general bounds on its Tur\'{a}n density are $\pi(H_k^r)…
In this paper we generalize the concept of uniquely $K_r$-saturated graphs to hypergraphs. Let $K_r^{(k)}$ denote the complete $k$-uniform hypergraph on $r$ vertices. For integers $k,r,n$ such that $2\le k <r<n$, a $k$-uniform hypergraph…
P. Erd\H{o}s [On extremal problems of graphs and generalized graphs, Israel Journal of Mathematics 2 (1964), 183-190] characterised those hypergraphs $F$ that have to appear in any sufficiently large hypergraph $H$ of positive density. We…
The {\em overlap number} of a finite $(d+1)$-uniform hypergraph $H$ is defined as the largest constant $c(H)\in (0,1]$ such that no matter how we map the vertices of $H$ into $\R^d$, there is a point covered by at least a $c(H)$-fraction of…
A $t$-dimensional orthogonal representation of a hypergraph is an assignment of nonzero vectors in $\mathbb{R}^t$ to its vertices, such that every hyperedge contains two vertices whose vectors are orthogonal. The orthogonality dimension of…
The Lagrangian density of an $r$-uniform hypergraph $F$ is $r!$ multiplying the supremum of the Lagrangians of all $F$-free $r$-uniform hypergraphs. For an $r$-graph $H$ with $t$ vertices, it is clear that $\pi_{\lambda}(H)\ge…
We show that $k$-ary functions giving the measure of the intersection of multi-parametric families of sets in probability spaces, e.g. $(x,y,z) \in X \times Y \times Z \mapsto \mu(P_{x,y} \cap Q_{x,z} \cap R_{y,z})$, satisfy a particularly…
Let $H=(V,E)$ be a hypergraph, where $V$ is a set of vertices and $E$ is a set of non-empty subsets of $V$ called edges. If all edges of $H$ have the same cardinality $r$, then $H$ is a $r$-uniform hypergraph; if $E$ consists of all…
We give lower bounds on the maximum possible girth of an $r$-uniform, $d$-regular hypergraph with at most $n$ vertices, using the definition of a hypergraph cycle due to Berge. These differ from the trivial upper bound by an absolute…
In this paper we asymptotically count $d$-regular $k$-uniform hypergraphs on $n$ vertices, provided $k$ is fixed and $d=d(n)=o(n^{1/2})$. In doing so, we extend to hypergraphs a switching technique of McKay and Wormald.